A bakery sells rolls in units of a dozen. The demand X (in 1000 units) for rolls has a gamma distribution with parameters α = 3, θ = 0.5, where θ is in units of days per 1000 units of rolls. It costs $ 2 to make a unit that sells for $ 5 on the first day when the rolls are fresh. Any leftover units are sold on the second day for $ 1. How many units should be made to maximize the expected value of the profit?

The value  is  

Step-by-step explanation:

From the question we are told that

   The  parameters  are  α = 3, θ = 0.5

    The cost of making a unit on the first day  is  c = $2

    The selling price of a  unit on the first day is  s = $5

    The selling price of a leftover unit on the second day is  v  = $ 1

Generally the profit of a unit on the first day is

The profit of a unit on the second day is

=>    

Generally the probability of making profit greater than $ 1 is mathematically represented as

=>  

Now from the gamma distribution table  we have that

Generally the probability of making profit less than or  equal to  $ 1 is mathematically represented as

=>    

=>        

So  the probability of making  $3  is    

and  the probability of making  -$1  is    

Generally the value of profit per day is mathematically represented as

=>    

=>